Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [720,2,Mod(163,720)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("720.163");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 720.z (of order \(4\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(5.74922894553\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(9\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 80) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 129 \nu^{17} + 124 \nu^{16} + 398 \nu^{15} - 116 \nu^{14} - 797 \nu^{13} - 2778 \nu^{12} + \cdots - 58624 ) / 1280 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 16 \nu^{17} - 19 \nu^{16} - 53 \nu^{15} + 2 \nu^{14} + 88 \nu^{13} + 331 \nu^{12} + 559 \nu^{11} + \cdots + 6048 ) / 160 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 75 \nu^{17} - 89 \nu^{16} - 248 \nu^{15} - 6 \nu^{14} + 375 \nu^{13} + 1487 \nu^{12} + 2550 \nu^{11} + \cdots + 28416 ) / 640 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 229 \nu^{17} - 258 \nu^{16} - 706 \nu^{15} + 120 \nu^{14} + 1377 \nu^{13} + 4800 \nu^{12} + \cdots + 89600 ) / 1280 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 178 \nu^{17} - 217 \nu^{16} - 614 \nu^{15} + 6 \nu^{14} + 954 \nu^{13} + 3793 \nu^{12} + \cdots + 75264 ) / 640 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 545 \nu^{17} + 684 \nu^{16} + 1918 \nu^{15} + 156 \nu^{14} - 2685 \nu^{13} - 11242 \nu^{12} + \cdots - 222976 ) / 1280 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 871 \nu^{17} + 1090 \nu^{16} + 3110 \nu^{15} + 352 \nu^{14} - 4043 \nu^{13} - 17564 \nu^{12} + \cdots - 351232 ) / 1280 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 228 \nu^{17} + 359 \nu^{16} + 1013 \nu^{15} + 672 \nu^{14} - 134 \nu^{13} - 3459 \nu^{12} + \cdots - 71232 ) / 320 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1017 \nu^{17} - 1536 \nu^{16} - 4342 \nu^{15} - 2508 \nu^{14} + 1381 \nu^{13} + 16526 \nu^{12} + \cdots + 345088 ) / 1280 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 129 \nu^{17} + 186 \nu^{16} + 524 \nu^{15} + 232 \nu^{14} - 321 \nu^{13} - 2280 \nu^{12} + \cdots - 46208 ) / 128 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 799 \nu^{17} - 1073 \nu^{16} - 3066 \nu^{15} - 930 \nu^{14} + 2727 \nu^{13} + 14975 \nu^{12} + \cdots + 305280 ) / 640 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1799 \nu^{17} + 2608 \nu^{16} + 7346 \nu^{15} + 3380 \nu^{14} - 4267 \nu^{13} - 31490 \nu^{12} + \cdots - 636160 ) / 1280 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 93 \nu^{17} + 133 \nu^{16} + 374 \nu^{15} + 162 \nu^{14} - 237 \nu^{13} - 1639 \nu^{12} - 3268 \nu^{11} + \cdots - 33024 ) / 64 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 981 \nu^{17} - 1344 \nu^{16} - 3828 \nu^{15} - 1348 \nu^{14} + 3013 \nu^{13} + 17886 \nu^{12} + \cdots + 363008 ) / 640 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1999 \nu^{17} + 2830 \nu^{16} + 8030 \nu^{15} + 3408 \nu^{14} - 5187 \nu^{13} - 35416 \nu^{12} + \cdots - 718848 ) / 1280 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 1039 \nu^{17} - 1484 \nu^{16} - 4198 \nu^{15} - 1844 \nu^{14} + 2547 \nu^{13} + 18238 \nu^{12} + \cdots + 372864 ) / 640 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 593 \nu^{17} - 812 \nu^{16} - 2294 \nu^{15} - 774 \nu^{14} + 1929 \nu^{13} + 10958 \nu^{12} + \cdots + 218944 ) / 320 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{16} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} - 2\beta_{11} + \beta_{8} + \beta_{4} - 2\beta_{3} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{4} - 2\beta_{3} + 2\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{17} - \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2 \beta_{17} + \beta_{16} - 5 \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{9} + 3 \beta_{8} + \cdots + 4 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{17} - 3 \beta_{16} + \beta_{15} + 2 \beta_{14} + \beta_{13} + 4 \beta_{12} + \beta_{11} + \cdots + 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2 \beta_{17} + \beta_{16} - 2 \beta_{14} - 7 \beta_{13} - \beta_{12} - 3 \beta_{11} + 11 \beta_{10} + \cdots - 6 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3 \beta_{16} - 11 \beta_{15} - \beta_{14} + 7 \beta_{13} + 12 \beta_{12} - 8 \beta_{10} + \cdots + 10 \beta_1 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 8 \beta_{17} + 4 \beta_{16} - 13 \beta_{15} - 7 \beta_{14} - 6 \beta_{13} - \beta_{12} + 3 \beta_{11} + \cdots + 22 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - \beta_{17} - 6 \beta_{16} - 6 \beta_{15} + 7 \beta_{14} + 4 \beta_{12} + 9 \beta_{11} - 3 \beta_{10} + \cdots + 24 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 18 \beta_{17} + 17 \beta_{16} + 5 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 10 \beta_{12} + 2 \beta_{10} + \cdots + 12 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 9 \beta_{17} - 21 \beta_{16} - 47 \beta_{15} - 20 \beta_{14} - 3 \beta_{13} + 34 \beta_{12} + \cdots - 28 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 14 \beta_{17} + 3 \beta_{16} - 24 \beta_{15} - 8 \beta_{14} - 3 \beta_{13} - 25 \beta_{12} + \cdots + 114 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 42 \beta_{17} + 25 \beta_{16} - 47 \beta_{15} - 7 \beta_{14} - 49 \beta_{13} + 72 \beta_{12} + \cdots + 90 \beta_1 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 32 \beta_{17} + 160 \beta_{16} - 75 \beta_{15} + 13 \beta_{14} + 104 \beta_{13} - 33 \beta_{12} + \cdots + 116 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 59 \beta_{17} - 196 \beta_{16} - 80 \beta_{15} - 103 \beta_{14} - 58 \beta_{13} + 82 \beta_{12} + \cdots + 8 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 54 \beta_{17} - 61 \beta_{16} - 63 \beta_{15} - 5 \beta_{14} - 121 \beta_{13} + 52 \beta_{12} + \cdots + 228 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).
| \(n\) | \(181\) | \(271\) | \(577\) | \(641\) |
| \(\chi(n)\) | \(\beta_{10}\) | \(-1\) | \(\beta_{10}\) | \(1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
−1.41267 | − | 0.0660953i | 0 | 1.99126 | + | 0.186742i | 2.00635 | − | 0.987189i | 0 | 1.55426 | − | 1.55426i | −2.80065 | − | 0.395417i | 0 | −2.89956 | + | 1.26196i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.2 | −1.19301 | + | 0.759419i | 0 | 0.846564 | − | 1.81200i | −2.17104 | + | 0.535339i | 0 | −2.13436 | + | 2.13436i | 0.366101 | + | 2.80463i | 0 | 2.18353 | − | 2.28740i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.3 | −0.828280 | − | 1.14628i | 0 | −0.627905 | + | 1.89888i | 0.245325 | + | 2.22257i | 0 | −0.343872 | + | 0.343872i | 2.69672 | − | 0.853049i | 0 | 2.34448 | − | 2.12212i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.4 | −0.567819 | + | 1.29521i | 0 | −1.35516 | − | 1.47090i | 1.42182 | − | 1.72581i | 0 | −1.60205 | + | 1.60205i | 2.67461 | − | 0.920026i | 0 | 1.42795 | + | 2.82151i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.5 | −0.430311 | − | 1.34716i | 0 | −1.62967 | + | 1.15939i | 0.177336 | − | 2.22902i | 0 | −0.115101 | + | 0.115101i | 2.26315 | + | 1.69652i | 0 | −3.07916 | + | 0.720273i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.6 | 0.516777 | − | 1.31641i | 0 | −1.46588 | − | 1.36058i | −2.07160 | − | 0.841703i | 0 | −1.13975 | + | 1.13975i | −2.54862 | + | 1.22659i | 0 | −2.17858 | + | 2.29211i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.7 | 1.23576 | + | 0.687667i | 0 | 1.05423 | + | 1.69959i | −0.832020 | − | 2.07551i | 0 | 2.83610 | − | 2.83610i | 0.134028 | + | 2.82525i | 0 | 0.399079 | − | 3.13699i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.8 | 1.29924 | − | 0.558542i | 0 | 1.37606 | − | 1.45136i | −1.49107 | + | 1.66635i | 0 | 2.40368 | − | 2.40368i | 0.977191 | − | 2.65426i | 0 | −1.00653 | + | 2.99782i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.9 | 1.38031 | − | 0.307817i | 0 | 1.81050 | − | 0.849763i | 1.71489 | + | 1.43498i | 0 | −0.458895 | + | 0.458895i | 2.23747 | − | 1.73024i | 0 | 2.80878 | + | 1.45284i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.1 | −1.41267 | + | 0.0660953i | 0 | 1.99126 | − | 0.186742i | 2.00635 | + | 0.987189i | 0 | 1.55426 | + | 1.55426i | −2.80065 | + | 0.395417i | 0 | −2.89956 | − | 1.26196i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.2 | −1.19301 | − | 0.759419i | 0 | 0.846564 | + | 1.81200i | −2.17104 | − | 0.535339i | 0 | −2.13436 | − | 2.13436i | 0.366101 | − | 2.80463i | 0 | 2.18353 | + | 2.28740i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.3 | −0.828280 | + | 1.14628i | 0 | −0.627905 | − | 1.89888i | 0.245325 | − | 2.22257i | 0 | −0.343872 | − | 0.343872i | 2.69672 | + | 0.853049i | 0 | 2.34448 | + | 2.12212i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.4 | −0.567819 | − | 1.29521i | 0 | −1.35516 | + | 1.47090i | 1.42182 | + | 1.72581i | 0 | −1.60205 | − | 1.60205i | 2.67461 | + | 0.920026i | 0 | 1.42795 | − | 2.82151i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.5 | −0.430311 | + | 1.34716i | 0 | −1.62967 | − | 1.15939i | 0.177336 | + | 2.22902i | 0 | −0.115101 | − | 0.115101i | 2.26315 | − | 1.69652i | 0 | −3.07916 | − | 0.720273i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.6 | 0.516777 | + | 1.31641i | 0 | −1.46588 | + | 1.36058i | −2.07160 | + | 0.841703i | 0 | −1.13975 | − | 1.13975i | −2.54862 | − | 1.22659i | 0 | −2.17858 | − | 2.29211i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.7 | 1.23576 | − | 0.687667i | 0 | 1.05423 | − | 1.69959i | −0.832020 | + | 2.07551i | 0 | 2.83610 | + | 2.83610i | 0.134028 | − | 2.82525i | 0 | 0.399079 | + | 3.13699i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.8 | 1.29924 | + | 0.558542i | 0 | 1.37606 | + | 1.45136i | −1.49107 | − | 1.66635i | 0 | 2.40368 | + | 2.40368i | 0.977191 | + | 2.65426i | 0 | −1.00653 | − | 2.99782i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.9 | 1.38031 | + | 0.307817i | 0 | 1.81050 | + | 0.849763i | 1.71489 | − | 1.43498i | 0 | −0.458895 | − | 0.458895i | 2.23747 | + | 1.73024i | 0 | 2.80878 | − | 1.45284i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.s | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 720.2.z.g | 18 | |
| 3.b | odd | 2 | 1 | 80.2.s.b | yes | 18 | |
| 5.c | odd | 4 | 1 | 720.2.bd.g | 18 | ||
| 12.b | even | 2 | 1 | 320.2.s.b | 18 | ||
| 15.d | odd | 2 | 1 | 400.2.s.d | 18 | ||
| 15.e | even | 4 | 1 | 80.2.j.b | ✓ | 18 | |
| 15.e | even | 4 | 1 | 400.2.j.d | 18 | ||
| 16.f | odd | 4 | 1 | 720.2.bd.g | 18 | ||
| 24.f | even | 2 | 1 | 640.2.s.c | 18 | ||
| 24.h | odd | 2 | 1 | 640.2.s.d | 18 | ||
| 48.i | odd | 4 | 1 | 320.2.j.b | 18 | ||
| 48.i | odd | 4 | 1 | 640.2.j.c | 18 | ||
| 48.k | even | 4 | 1 | 80.2.j.b | ✓ | 18 | |
| 48.k | even | 4 | 1 | 640.2.j.d | 18 | ||
| 60.h | even | 2 | 1 | 1600.2.s.d | 18 | ||
| 60.l | odd | 4 | 1 | 320.2.j.b | 18 | ||
| 60.l | odd | 4 | 1 | 1600.2.j.d | 18 | ||
| 80.s | even | 4 | 1 | inner | 720.2.z.g | 18 | |
| 120.q | odd | 4 | 1 | 640.2.j.c | 18 | ||
| 120.w | even | 4 | 1 | 640.2.j.d | 18 | ||
| 240.t | even | 4 | 1 | 400.2.j.d | 18 | ||
| 240.z | odd | 4 | 1 | 80.2.s.b | yes | 18 | |
| 240.bb | even | 4 | 1 | 320.2.s.b | 18 | ||
| 240.bd | odd | 4 | 1 | 400.2.s.d | 18 | ||
| 240.bd | odd | 4 | 1 | 640.2.s.d | 18 | ||
| 240.bf | even | 4 | 1 | 640.2.s.c | 18 | ||
| 240.bf | even | 4 | 1 | 1600.2.s.d | 18 | ||
| 240.bm | odd | 4 | 1 | 1600.2.j.d | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 80.2.j.b | ✓ | 18 | 15.e | even | 4 | 1 | |
| 80.2.j.b | ✓ | 18 | 48.k | even | 4 | 1 | |
| 80.2.s.b | yes | 18 | 3.b | odd | 2 | 1 | |
| 80.2.s.b | yes | 18 | 240.z | odd | 4 | 1 | |
| 320.2.j.b | 18 | 48.i | odd | 4 | 1 | ||
| 320.2.j.b | 18 | 60.l | odd | 4 | 1 | ||
| 320.2.s.b | 18 | 12.b | even | 2 | 1 | ||
| 320.2.s.b | 18 | 240.bb | even | 4 | 1 | ||
| 400.2.j.d | 18 | 15.e | even | 4 | 1 | ||
| 400.2.j.d | 18 | 240.t | even | 4 | 1 | ||
| 400.2.s.d | 18 | 15.d | odd | 2 | 1 | ||
| 400.2.s.d | 18 | 240.bd | odd | 4 | 1 | ||
| 640.2.j.c | 18 | 48.i | odd | 4 | 1 | ||
| 640.2.j.c | 18 | 120.q | odd | 4 | 1 | ||
| 640.2.j.d | 18 | 48.k | even | 4 | 1 | ||
| 640.2.j.d | 18 | 120.w | even | 4 | 1 | ||
| 640.2.s.c | 18 | 24.f | even | 2 | 1 | ||
| 640.2.s.c | 18 | 240.bf | even | 4 | 1 | ||
| 640.2.s.d | 18 | 24.h | odd | 2 | 1 | ||
| 640.2.s.d | 18 | 240.bd | odd | 4 | 1 | ||
| 720.2.z.g | 18 | 1.a | even | 1 | 1 | trivial | |
| 720.2.z.g | 18 | 80.s | even | 4 | 1 | inner | |
| 720.2.bd.g | 18 | 5.c | odd | 4 | 1 | ||
| 720.2.bd.g | 18 | 16.f | odd | 4 | 1 | ||
| 1600.2.j.d | 18 | 60.l | odd | 4 | 1 | ||
| 1600.2.j.d | 18 | 240.bm | odd | 4 | 1 | ||
| 1600.2.s.d | 18 | 60.h | even | 2 | 1 | ||
| 1600.2.s.d | 18 | 240.bf | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):
|
\( T_{7}^{18} - 2 T_{7}^{17} + 2 T_{7}^{16} + 32 T_{7}^{15} + 200 T_{7}^{14} - 120 T_{7}^{13} + 352 T_{7}^{12} + \cdots + 288 \)
|
|
\( T_{11}^{18} - 2 T_{11}^{17} + 2 T_{11}^{16} + 80 T_{11}^{15} + 848 T_{11}^{14} + 64 T_{11}^{13} + \cdots + 5431808 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} - 2 T^{16} + \cdots + 512 \)
|
| $3$ |
\( T^{18} \)
|
| $5$ |
\( T^{18} + 2 T^{17} + \cdots + 1953125 \)
|
| $7$ |
\( T^{18} - 2 T^{17} + \cdots + 288 \)
|
| $11$ |
\( T^{18} - 2 T^{17} + \cdots + 5431808 \)
|
| $13$ |
\( T^{18} + 112 T^{16} + \cdots + 67108864 \)
|
| $17$ |
\( T^{18} - 6 T^{17} + \cdots + 512 \)
|
| $19$ |
\( T^{18} + 2 T^{17} + \cdots + 4608 \)
|
| $23$ |
\( T^{18} + \cdots + 17700587552 \)
|
| $29$ |
\( T^{18} + 14 T^{17} + \cdots + 82330112 \)
|
| $31$ |
\( T^{18} + 196 T^{16} + \cdots + 16384 \)
|
| $37$ |
\( T^{18} + \cdots + 574297214976 \)
|
| $41$ |
\( T^{18} + \cdots + 242788765696 \)
|
| $43$ |
\( T^{18} + \cdots + 337207844416 \)
|
| $47$ |
\( T^{18} + \cdots + 16870640672 \)
|
| $53$ |
\( (T^{9} + 6 T^{8} + \cdots - 220832)^{2} \)
|
| $59$ |
\( T^{18} + \cdots + 144166720393728 \)
|
| $61$ |
\( T^{18} + \cdots + 121236758528 \)
|
| $67$ |
\( T^{18} + \cdots + 555525752896 \)
|
| $71$ |
\( (T^{9} + 12 T^{8} + \cdots + 27648)^{2} \)
|
| $73$ |
\( T^{18} + \cdots + 35535647232 \)
|
| $79$ |
\( (T^{9} + 8 T^{8} + \cdots + 45002752)^{2} \)
|
| $83$ |
\( (T^{9} + 20 T^{8} + \cdots + 8413744)^{2} \)
|
| $89$ |
\( (T^{9} + 6 T^{8} + \cdots + 251904)^{2} \)
|
| $97$ |
\( T^{18} + \cdots + 380349381734912 \)
|
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